A nonhomogeneous Markov process for the estimation of Gaussian random fields with nonlinear observations. (English) Zbl 0739.60040

The authors consider the following estimation problem: the signal is modelled by an \(n\)-dimensional continuous Gaussian field on a bounded and smooth domain in \(R^ d\), whose covariance is the Green’s tensor of some uniformly elliptic differential system. This signal is observed through a finite set of bounded and smooth nonlinear sensors corrupted by noise. As usual, it is necessary to compute the conditional field, but the main difficulty is that its distribution is not Gaussian. To overcome this difficulty the authors propose the following way: they work with an increasing sequence of finite-dimensional subspaces of the space of continuous functions, on each of which a Galerkin approximation to the conditional field is considered. Then they construct a special nonhomogeneous Markov process which passes through this sequence of subspaces in time. This Markov process converges in distribution to the conditional field and the empirical averages of any bounded measurable function, continuous a.s., converge in \(L_ 2\) to the conditional expectation of the function.


60G60 Random fields
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
60J25 Continuous-time Markov processes on general state spaces
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